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Potential Analysis

, Volume 37, Issue 2, pp 125–169 | Cite as

On a Class of Markov Semigroups on Discrete Ultra-Metric Spaces

  • Alexander Bendikov
  • Alexander Grigor’yanEmail author
  • Christophe Pittet
Article

Abstract

We consider a discrete ultra-metric space \(\left( X,d\right) \) with a measure m and construct in a natural way a symmetric Markov semigroup \( \left\{ P_{t}\right\} _{t\geq 0}\) in \(L^{2}\left( X,m\right) \) and the corresponding Markov process \(\left\{ \mathcal{X}_{t}\right\} \). We prove upper and lower bounds of its transition function and its Green function, give a criterion for the transience, and estimate its moments.

Keywords

Markov chain Markov semigroup Markov generator Ultra-metric space Heat kernel Transition probability 

Mathematics Subject Classifications (2010)

Primary 60J05; Secondary 60J27 60J35 60B15 

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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Alexander Bendikov
    • 1
  • Alexander Grigor’yan
    • 2
    Email author
  • Christophe Pittet
    • 3
  1. 1.Institute of MathematicsWrocław UniversityWrocławPoland
  2. 2.Department of MathematicsUniversity of BielefeldBielefeldGermany
  3. 3.CMIUniversité d’Aix-Marseille IMarseille Cedex 13France

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